Math Fun

For discussing anything related to physics, biology, chemistry, mathematics, and their practical applications.

Moderator: Flannel Jesus

Re: Math Fun

Sauwelios wrote:
Fixed Cross wrote:y'all copied the canonical and began defending it at a point where it wasn't even being attacked, but all failed, even refused to demonstrate the epistemic necessity of the guru (which is the point you're relying on)

This is just an ungrounded insinuation. Are you in fact trolling here?

It's my honest impression. Should I accuse you of trolling every time you fail to be convinced by some point I want to make?

I did penetrate that, first by thinking that the hypothesis "if there was only one and the guru said she sa[w] one" was false. And it still is.

The "if" part of an "if/then" statement (jargon?) cannot be true or false, as it's not a complete statement...

It is falsely inserted.

Anyway, your reasoning seems to go like this:

1. The one, blue-eyed person does not know his own eye colour.
2. The guru tells him that she sees a blue-eyed person.
3. The one, blue-eyed person, supposing that the guru is to be believed, knows he has blue eyes.

Now according to you, it seems, the 3rd step somehow retroactively cancels the 1st. This is nonsense.

"the blue eyed person does not know his eye color"
"the blue eyed person does know his eye color"

Yes, these statements contradict each other.

You (all) seem to think that "Supposing that the guru [speaks and] is to be believed" applies only to step 3. Where, if it is valid at 3., it would also have been valid at 1. which cancels out the content of 1.

Oh wait, I am thinking.
I mean rambling.

By the by, as I've shown (and as Sauwelios even admitted when I told him "humbly" (the key to teaching) the explicit text doesn't even allow for the canonical solution, as all people see each other at all times, disallowing for the whole "in the morning they were still there" scenario.

I didn't "admit" that, it's just a truism.

It's simply true. But not to the point here.

Perhaps it's because you don't think in words, that you can't put your thoughts into words very well.

Abstract thoughts, when they are in fact a force, insight, always are hard to put into words. Clever word-play is often mistaken for thinking. I'm not of that school, indeed.
The strong do what they can, the weak accept what they must.
- Thucydides
-
Before the Light - Philosophy 77 - sumofalltemples - The Magickal Tree of Life Academy

Fixed Cross
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Re: Math Fun

Flannel Jesus wrote:I personally doubt that there's any single proof which covers both.

I personally doubt that there isn't. A proof of a biconditional like "the problem is solvable IFF the board has [some property or set of properties]" would necessarily prove both.

On a much smaller board, the mutual solution is easy: A two by two board is solvable if and only if the removed squares are adjacent, because that implies that the remaining squares are adjacent, and the remaining 2-square board is solvable if and only if it is a 2x1 rectangle.

I would guess that adjacency is a special case of some property for which color is a proxy, and that "the board is solvable IFF the removed squares have property X such that X is a superset of adjacency".
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Carleas
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Re: Math Fun

Sauwelios wrote:I demonstrated the false steps in his reasoning in that post of mine in which I made some phrases blue and some red. James need not concur: he has not shown how his reasoning is impervious to my critique.

Oh really?
I'm listening...

"IF they all start counting from the same number, will they always be able to deduce their eye color?"
Given: They ALL start with the SAME number for counting in the following way;
1) they count the number of blues they see above the number they started counting from, the "97" or "1" or whatever.
2) they count the days of no one leaving.
3) if they count less blues above the start number to the count of non-leaving days, they deduce they are the last blue.

Try it all you want with any number. It will always work to separate the browns from the blues.
Try to find any number that doesn't work through that process.
Clarify, Verify, Instill, and Reinforce the Perception of Hopes and Threats unto Anentropic Harmony
Else
From THIS age of sleep, Homo-sapien shall never awake.

The Wise gather together to help one another in EVERY aspect of living.

You are always more insecure than you think, just not by what you think.
The only absolute certainty is formed by the absolute lack of alternatives.
It is not merely "do what works", but "to accomplish what purpose in what time frame at what cost".
As long as the authority is secretive, the population will be subjugated.

Amid the lack of certainty, put faith in the wiser to believe.
Devil's Motto: Make it look good, safe, innocent, and wise.. until it is too late to choose otherwise.

The Real God ≡ The reason/cause for the Universe being what it is = "The situation cannot be what it is and also remain as it is".
.
James S Saint
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Re: Math Fun

I found that much less annoying than how you normally write in this thread. It seems at least far less arrogant and oblivious.

Silhouette wrote:Please start by explaining why the Guru "saying she sees one" is incompatible with the thought experiment of "what one would do if there was only one".

Jesus, just fucking read my posts. My God. Or better yet, think.

As for the "all people see each other at all times" thing - this is not incompatible with anyone leaving or staying on the island (perhaps the ferry doesn't take them out of view, and they might even see perfectly well at night and never sleep for all we know). The whole "see them in the morning" thing is perfectly fine when taken as it was intended, as "seen on the island even after the ferry had been and gone, picking up anyone who knew their eye colour while it was there at midnight - as in the puzzle". Even this attempt at a minor point seems to hold no water.

Right. Sau says it's a truism, you say it's false -
you guys are clowns.

And please, to both you and Sau, stop squabbling. It really doesn't lend yourself any of the credibility either of you are fighting for here.

Sillybilly you are a true comedian.
The strong do what they can, the weak accept what they must.
- Thucydides
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Before the Light - Philosophy 77 - sumofalltemples - The Magickal Tree of Life Academy

Fixed Cross
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Re: Math Fun

Fixed Cross wrote:Sau says it's a truism, you say it's false

This does not suggest that either is wrong, but that the problem as originally stated is ambiguous. I think your clarification, that the islanders can only see each other during the day, and they disappear during the night if they learn their eye color, resolves the ambiguity.
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Carleas
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Re: Math Fun

Fixed Cross wrote:I found that much less annoying than how you normally write in this thread. It seems at least far less arrogant and oblivious.

Right. So saying "just fucking read my posts. My God. Or better yet, think" as though your posts are so obvious (even though nobody gets them), and that *I'm* the one who isn't thinking even though what I've said has been agreed with on multiple occasions, and even referenced in order to help combat the two trailing contributors who still don't get it.

What you say doesn't make any more sense now than it did the first time I read it. Take a leaf out of my book - from the part where I have proactively improved on my original presentation of the correct solution on multiple occasions, just to more clearly rebuke all the presented criticisms of it and how it is the only correct solution. That's how you gain credibility: proactively showing and improving on the many hours of thought you've put into solving a problem and problems with others not understanding it. And when they don't, try again and harder until they do.

Like I said: your lazy approach of merely claiming correctness without (clear) demonstration - whilst only angrily pointing fingers at others for not being as clever as you think you are "to back you up" will get you nowhere.

Silhouette
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Re: Math Fun

Fixed Cross wrote:
Anyway, your reasoning seems to go like this:

1. The one, blue-eyed person does not know his own eye colour.
2. The guru tells him that she sees a blue-eyed person.
3. The one, blue-eyed person, supposing that the guru is to be believed, knows he has blue eyes.

Now according to you, it seems, the 3rd step somehow retroactively cancels the 1st. This is nonsense.

"the blue eyed person does not know his eye color"
"the blue eyed person does know his eye color"

Yes, these statements contradict each other.

You (all) seem to think that "Supposing that the guru [speaks and] is to be believed" applies only to step 3. Where, if it is valid at 3., it would also have been valid at 1. which cancels out the content of 1.

Er, have the 9-11 attacks cancelled out the fact that the Twin Towers ever existed?
"Someone may object that the successful revolt against the universal and homogeneous state could have no other effect than that the identical historical process which has led from the primitive horde to the final state will be repeated. But would such a repetition of the process--a new lease of life for man's humanity--not be preferable to the indefinite continuation of the inhuman end? Do we not enjoy every spring although we know the cycle of the seasons, although we know that winter will come again?" (Leo Strauss, "Restatement on Xenophon's Hiero".)

Sauwelios
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Re: Math Fun

I think sau hit the nail on the head: perhaps FC didn't realize that 3 was taking place after 1? 1, 2, and 3 were a chronological sequence of facts/events.

Flannel Jesus

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Re: Math Fun

????
Did the towers need to fall hypothetically after a guru said they had fallen before someone could calculate that they had actually fallen?

You don't seem to understand that the hypothesis they're using is not occurring within the same timeline as the actual events, but in their mind.
The strong do what they can, the weak accept what they must.
- Thucydides
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Fixed Cross
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Re: Math Fun

Silhouette wrote:
Fixed Cross wrote:I found that much less annoying than how you normally write in this thread. It seems at least far less arrogant and oblivious.

Right. So saying "just fucking read my posts. My God. Or better yet, think" as though your posts are so obvious (even though nobody gets them), and that *I'm* the one who isn't thinking even though what I've said has been agreed with on multiple occasions, and even referenced in order to help combat the two trailing contributors who still don't get it.

What you say doesn't make any more sense now than it did the first time I read it. Take a leaf out of my book - from the part where I have proactively improved on my original presentation of the correct solution on multiple occasions, just to more clearly rebuke all the presented criticisms of it and how it is the only correct solution. That's how you gain credibility: proactively showing and improving on the many hours of thought you've put into solving a problem and problems with others not understanding it. And when they don't, try again and harder until they do.

Like I said: your lazy approach of merely claiming correctness without (clear) demonstration - whilst only angrily pointing fingers at others for not being as clever as you think you are "to back you up" will get you nowhere.

After about a fucking million time I told you that I GET THE ORIGINAL SOLUTION.

Honestly, dude, if even that is too difficult for you to understand -- that I get the original solution, as I've repeated and repeated an repeated -- it is not much of a miracle that you don't understand my posts. But I don't feel that I can do anything about that.
Last edited by Fixed Cross on Fri May 17, 2013 10:12 pm, edited 1 time in total.
The strong do what they can, the weak accept what they must.
- Thucydides
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Before the Light - Philosophy 77 - sumofalltemples - The Magickal Tree of Life Academy

Fixed Cross
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Re: Math Fun

There are four blues and four browns.
All know that all see that there is at least one of each color: The REAL SET of REAL UNITS.
The day/night/ferry things apply.

All reason:
- If there was only one blue eyed who knew that there was at least one blue eyed (one REAL UNIT of the REAL SET), he would leave the first day. And:
- If there was only one brown eyed who knew that there was at least one brown eyed (one REAL UNIT of the REAL SET), he would leave the first day.

No one leaves the first day. Therefore there are more than 1 of each color, and all know that they might belong to either one, or possibly another color.

Since no one leaves the second day, there are more than 2 of each color.
Since no one leaves the third say, there are more than 3 of each color.
Since all blues see only 3 blues, they leave the 4th day. Since all browns see only 3 browns, they all leave the 4th day.
The strong do what they can, the weak accept what they must.
- Thucydides
-
Before the Light - Philosophy 77 - sumofalltemples - The Magickal Tree of Life Academy

Fixed Cross
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Re: Math Fun

Wow, FC, you take some serious time out from this place. A good thing. Well, two weeks.... I'd hoped it was due to bowing out after realising your error that Sau so sufficiently presented in analogous form, but I guess that was wishful thinking.

Fixed Cross wrote:Did the towers need to fall hypothetically after a guru said they had fallen before someone could calculate that they had actually fallen?

You know that's not an argument, right?
And that it does nothing to defend your strange claim that someone can not be ignorant of something one moment, come to know it, and know it the next moment. Somehow, to you, this is a contradiction - that if they knew it the next moment, they must have known it all along. Yes, that's what it sounds like you are saying.

Fixed Cross wrote:After about a fucking million time I told you that I GET THE ORIGINAL SOLUTION.

Ok. Whilst I hear you say such things, things like this say otherwise:

Fixed Cross wrote:There are four blues and four browns.
All know that all see that there is at least one of each color: The REAL SET of REAL UNITS.
The day/night/ferry things apply.

All reason:
- If there was only one blue eyed who knew that there was at least one blue eyed (one REAL UNIT of the REAL SET), he would leave the first day. And:
- If there was only one brown eyed who knew that there was at least one brown eyed (one REAL UNIT of the REAL SET), he would leave the first day.

No one leaves the first day. Therefore there are more than 1 of each color, and all know that they might belong to either one, or possibly another color.

Since no one leaves the second day, there are more than 2 of each color.
Since no one leaves the third say, there are more than 3 of each color.
Since all blues see only 3 blues, they leave the 4th day. Since all browns see only 3 browns, they all leave the 4th day.

As I thought, and have said before, your mistake is to say that since 4 blues (or browns) know that everyone else knows that everyone else knows that there is at least 1 blue-eyed islander and at least 1 brown-eyed islander, 1 blue (or brown) must also know that everyone else knows that everyone else knows that there are at least 1 blue-eyed islander and at least 1 brown-eyed islander.

I know why you think this is an ok assumption, because the reality (in this 4blue/brown scenario) is that there ARE 4 blues and 4 browns, and so they can attribute their knowledge to whatever scanerios they are imagining. The problem is that they have to be hypothesising about what 1 blue/brown would know IF there were only 1 blue/brown. In this case, they have to deduce about this 1 blue/brown aside from the knowledge of 4 blues/browns.

The reason you don't get this appears to be at least one reason why you don't get the correct solution. You can say you get the solution a million more times, but when you consistently demonstrate that you do not, it just holds no water. You have to do better than just "saying" you get it.

Fixed Cross wrote:Honestly, dude, if even that is too difficult for you to understand -- that I get the original solution, as I've repeated and repeated an repeated -- it is not much of a miracle that you don't understand my posts. But I don't feel that I can do anything about that.

Perhaps it is too difficult for you to understand what I do. It sounds like we both think the same of the other in regard to the validity of one another's positions. At least one of us has the correct understanding of the problem. How will we decide who this person is? It would be wrong to base my validity on the fact that all other members of this place and the place that hosted the problem side with my explanations, bar the one "I want to be special, even at the expense of sense" person here, and yourself. We must arrive at a much more acceptable measure of validity - any suggestions? Showing you flawless reasoning doesn't appear to be sufficient so far. I will continue to try.

Silhouette
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Re: Math Fun

Silhouette wrote:At least one of us has the correct understanding of the problem. How will we decide who this person is?

This question is really the only reason I keep discussing this. It is possibly the most important question I've come across in my years of discussing philosophy.

I think the answer starts this way: what would it take to prove you that you're wrong? For me, I think it would be a logical proof, using explicitly and exhaustively stated premises. I think that I can point to an implied premise that is false, and that thus if all the premises were explicitly and exhaustively stated and did not rely on the implied premise I see, I would be proven wrong.

Fixed Cross, I have a thought for you: You say that, "If there was only one blue eyed who knew that there was at least one blue eyed (one REAL UNIT of the REAL SET), he would leave the first day," and we don't agree. But it is also true that "If there was only one blue, he could not learn that there is at least one blue by looking at the other islanders' eyes." And yet, every person on the island knows "there is at least one blue" only because they can see that by looking at the other islanders' eyes. This distinguishes the knowledge the islanders have before and after the guru speaks: if there were only one blue, he could learn that there is at least one blue by hearing the guru speak.
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Carleas
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Re: Math Fun

Carleas wrote:
Silhouette wrote:At least one of us has the correct understanding of the problem. How will we decide who this person is?

This question is really the only reason I keep discussing this. It is possibly the most important question I've come across in my years of discussing philosophy.

I think the answer starts this way: what would it take to prove you that you're wrong? For me, I think it would be a logical proof, using explicitly and exhaustively stated premises. I think that I can point to an implied premise that is false, and that thus if all the premises were explicitly and exhaustively stated and did not rely on the implied premise I see, I would be proven wrong.

That is your mistake. You fail at that again and again.
The only proof is one that leaves the absolute lack of alternatives.
The canonical solution is "A solution". But it is NOT the "Only solution", and thus is not a proof of anything else being wrong.
You constantly attempt to shoot down a premise by presuming where it is going to lead and thus claim it to be invalid before finding out where it actually leads. You conflate issues in the effort to prevent the beginning of a scenario. That is pure political strategy, not logic.

In order to prove anyone else wrong, you MUST go through their scenario and point out at exactly what point it fails or prove the absolute lack of alternatives to a different argument. But to do that, you must first except the proposed premises that you might LATER shoot down as inapplicable to the final issue. But you refuse to do that. You want to jump ahead and presume your own conclusion; "I think this premise will be a problem later so I am not going to accept it now". The problem is that you don't actually know for certain whether it will be or not and refuse to find out. You are "affirming a presumed consequent".

"I won't accept the premise of evolution for discussion because that would mean there is no God."
...Bullshit.
"I refuse to discuss the whereabouts of the defendant because I already know who is guilty."
...Contempt of court.
Clarify, Verify, Instill, and Reinforce the Perception of Hopes and Threats unto Anentropic Harmony
Else
From THIS age of sleep, Homo-sapien shall never awake.

The Wise gather together to help one another in EVERY aspect of living.

You are always more insecure than you think, just not by what you think.
The only absolute certainty is formed by the absolute lack of alternatives.
It is not merely "do what works", but "to accomplish what purpose in what time frame at what cost".
As long as the authority is secretive, the population will be subjugated.

Amid the lack of certainty, put faith in the wiser to believe.
Devil's Motto: Make it look good, safe, innocent, and wise.. until it is too late to choose otherwise.

The Real God ≡ The reason/cause for the Universe being what it is = "The situation cannot be what it is and also remain as it is".
.
James S Saint
ILP Legend

Posts: 25976
Joined: Sun Apr 18, 2010 8:05 pm

Re: Math Fun

1) If I prove a statement A, I know as a corollary that ~~A. If I prove a collection of statements A and A -> B, I know as a corollary that B. If a rigorous logical process leads from a collection of premises to a conclusion, I know that the negation of that conclusion can only be true if the premises produce a contradiction. Since you haven't offered anything to undermine the logic of the canonical solution, the best you're shooting for is proving the givens to be contradictory.

2) If I say "Here are the givens. Teddy bear. Therefore, solution," have I really proved anything? How much do you need to "go through the[] scenario and point out at exactly what point it fails or prove the absolute lack of alternatives to a different argument"? Your solution has everyone pick the same number from thin air (unless they have brown eyes), turn that number by magic into knowledge, and then deduce from that knowledge. I'll be honest, I don't know how to explain logically why the color of my eyes doesn't follow from the number we all start counting at. It's a non sequitur. The one thing doesn't follow from the other.

3) Related to point #2, you did not identify what would convince you that you are wrong.
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Carleas
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Re: Math Fun

You're drunk, right?
Clarify, Verify, Instill, and Reinforce the Perception of Hopes and Threats unto Anentropic Harmony
Else
From THIS age of sleep, Homo-sapien shall never awake.

The Wise gather together to help one another in EVERY aspect of living.

You are always more insecure than you think, just not by what you think.
The only absolute certainty is formed by the absolute lack of alternatives.
It is not merely "do what works", but "to accomplish what purpose in what time frame at what cost".
As long as the authority is secretive, the population will be subjugated.

Amid the lack of certainty, put faith in the wiser to believe.
Devil's Motto: Make it look good, safe, innocent, and wise.. until it is too late to choose otherwise.

The Real God ≡ The reason/cause for the Universe being what it is = "The situation cannot be what it is and also remain as it is".
.
James S Saint
ILP Legend

Posts: 25976
Joined: Sun Apr 18, 2010 8:05 pm

Re: Math Fun

James, insults, 3-word dismissals, and repeating the same, oft-answered question have not made your position any stronger.
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Carleas
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Re: Math Fun

Just to repeat:
the lack of alternatives has been covered by the exhaustive nature of correctly following the logic of the correct solution.

To every blue there are ONLY two possibilities: there are 100 blues or 99. The rest of the solution is examining each of these outcomes just as exhaustively, leaving no stone unturned, and leading to only one solution. It works out very nicely if you really do understand it.

Silhouette
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Re: Math Fun

Carleas wrote:James, insults, 3-word dismissals, and repeating the same, oft-answered question have not made your position any stronger.

You refusing to discuss anything but your own preferred scenario is not helping your case either (and you haven't answered the question at all... always trying to jump around it like a frog on a hot plate).
Silhouette wrote:Just to repeat:
the lack of alternatives has been covered by the exhaustive nature of correctly following the logic of the correct solution.

That is absurd.

The canonical solution depends, not only on everyone knowing that everyone knows all of those things we listed, but it also depends on everyone knowing that everyone has no alternative but to be thinking exactly the same scenario.

The problem is that there is a different scenario that also works. So how do they all know that all of the others are using the canonical solution rather than another? Does the guru also tell them that there is one person thinking, "If I was brown and saw 99 blues, then...". That part wasn't mentioned.

You are each merely following a proscribed trail. There are alternatives. And because of that, they cannot each know that all of the others are waiting for the same reason.
Clarify, Verify, Instill, and Reinforce the Perception of Hopes and Threats unto Anentropic Harmony
Else
From THIS age of sleep, Homo-sapien shall never awake.

The Wise gather together to help one another in EVERY aspect of living.

You are always more insecure than you think, just not by what you think.
The only absolute certainty is formed by the absolute lack of alternatives.
It is not merely "do what works", but "to accomplish what purpose in what time frame at what cost".
As long as the authority is secretive, the population will be subjugated.

Amid the lack of certainty, put faith in the wiser to believe.
Devil's Motto: Make it look good, safe, innocent, and wise.. until it is too late to choose otherwise.

The Real God ≡ The reason/cause for the Universe being what it is = "The situation cannot be what it is and also remain as it is".
.
James S Saint
ILP Legend

Posts: 25976
Joined: Sun Apr 18, 2010 8:05 pm

Re: Math Fun

James S Saint wrote:The canonical solution depends, not only on everyone knowing that everyone knows all of those things we listed, but it also depends on everyone knowing that everyone has no alternative but to be thinking exactly the same scenario.

This is false. The canonical solution is not about what everyone is thinking but about what they can deduce. It is a given that if something can be logically deduced, every islander will deduce it instantly, and the islanders know this about each other. There is no choice of "alternatives": if we are given A and A -> B, and we are perfect logicians who will instantly deduce anything deducible, we will instantly deduce B.
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Carleas
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Re: Math Fun

Carleas wrote:
James S Saint wrote:The canonical solution depends, not only on everyone knowing that everyone knows all of those things we listed, but it also depends on everyone knowing that everyone has no alternative but to be thinking exactly the same scenario.

This is false. The canonical solution is not about what everyone is thinking but about what they can deduce. It is a given that if something can be logically deduced, every islander will deduce it instantly, and the islanders know this about each other. There is no choice of "alternatives": if we are given A and A -> B, and we are perfect logicians who will instantly deduce anything deducible, we will instantly deduce B.

Maybe you should reread your solution... "If number 99 is thinking... that number 98 is thinking...".
Clarify, Verify, Instill, and Reinforce the Perception of Hopes and Threats unto Anentropic Harmony
Else
From THIS age of sleep, Homo-sapien shall never awake.

The Wise gather together to help one another in EVERY aspect of living.

You are always more insecure than you think, just not by what you think.
The only absolute certainty is formed by the absolute lack of alternatives.
It is not merely "do what works", but "to accomplish what purpose in what time frame at what cost".
As long as the authority is secretive, the population will be subjugated.

Amid the lack of certainty, put faith in the wiser to believe.
Devil's Motto: Make it look good, safe, innocent, and wise.. until it is too late to choose otherwise.

The Real God ≡ The reason/cause for the Universe being what it is = "The situation cannot be what it is and also remain as it is".
.
James S Saint
ILP Legend

Posts: 25976
Joined: Sun Apr 18, 2010 8:05 pm

Re: Math Fun

James S Saint wrote:The canonical solution depends, not only on everyone knowing that everyone knows all of those things we listed, but it also depends on everyone knowing that everyone has no alternative but to be thinking exactly the same scenario.

Consider 2+2=4.

As long as everyone accepts the logical definitions of 2, +, = and 4, they are necessarily going to agree with this equation.
If the scenario is that there are "2 here" and "2 there" and nothing besides, and if the sum of those things equals 4, then everyone on an island leaves the island on a ferry at midnight (otherwise nothing changes) then the only things that have any consequence are 2, +, = and 4, and the logic that 2+2=4.

Given that everyone involved is a perfect logician and thus accepts logical definitions, and everyone is aware of the limited number of things that have any consequence, you can know that everyone is ONLY going to be considering these things within these limits. In the above scenario, 2+2=4 is non-negotiable, and the ONLY thing of consequence, so you can KNOW everyone is going to be deducing this in this way, and nothing else. There is no worry of anyone thinking differently.

_________________________

Now, in the actual puzzle, blue-eyed islanders are ONLY going to be considering the consequences of whether there are 100 blue-eyed islanders or 99. They see 99 and do not know whether they themselves have blue eyes - if they did there would be 100, if not there would be 99. So there are ONLY 2 possibilities to explore:
1) The consequence of there being 100 is that each blue-eyed islander knows they have blue eyes, and leave on the ferry in the knowledge that they have blue eyes.
2) The consequence of there being 99 is that 99 blue-eyed islanders would be ONLY considering the consequences of whether there are 99 blue-eyed islanders or 98. They would see 98 and not know whether they themselves have blue eyes - if they did there would be 99, if not there would be 98. So there would be ONLY 2 possibilities to explore:
......1) The consequence of there being 99 is that each blue-eyed islander knows they have blue eyes, and leave on the ferry in the knowledge that they
......have blue eyes.
......2) The consequence of there being 98 is that 98 blue-eyed islanders would be ONLY considering the consequences of whether there are 98 blue-eyed
......islanders or 97. They would see 97 and not know whether they themselves have blue eyes - if they did there would be 98, if not there would be 97. So
......there are ONLY 2 possibilities to explore:
............1) The consequences of there being 98...., and
............2) The consequences of there being 97.... this is recursive....
Until:
1) The consequences of there being 1: that they would leave on the ferry in the knowledge that they have blue eyes, and
2) The consequences of there being 0.... BUT THERE ARE NOT 0! - the Guru revealed otherwise.

So it's finally certain that if there were 1, they would leave at the next opportunity. They don't so there aren't one. We move back up to:
1) The consequences of there being 2: that they would leave on the ferry at the NEXT opportunity in the knowledge that they have blue eyes, and
2) The consequences of there being 1.... BUT THERE ARE NOT 1! - nobody left at the FIRST opportunity.

This is also recursive. It leads back to there definitely being 100 on day 100, and thus all the blues finally know they have blue eyes... AND ALL FROM EXAMINING THE ONLY POSSIBILITIES. There were no other possibilities that were missed out here. And all possibilities were examined.
THIS is why there are no alternatives to the correct solution.

This is the best way in which I've explained it yet.

Silhouette
Philosopher

Posts: 4511
Joined: Tue May 20, 2003 1:27 am
Location: Existence

Re: Math Fun

You guys keep trying to reduce the "perfect logician" to being no more than a calculator wherein someone has to push his buttons. In the case of the ultra android, when someone pushes his 2+2 button sequence, it might well reach out slap him as well as add the numbers.

As soon as anyone mentions a rule wherein someone must leave if they deduce their eye color, every perfect logician would instantly deduce what would be required for that to happen, whether they wanted it to or not. That means that they would instantly become aware of every logical scenario that would lead to such a deduction, not merely any one particular scenario that would depend on someone pushing their buttons. And all of them would be thinking about all possible scenarios.

One of the scenarios that would allow the deduction of their color would be dependent upon a guru saying that there was at least one blue and everyone hearing and believing that. But before the guru even had a chance to say that, they would all already know of a scenario that allowed such a deduction without the guru speaking at all, as well as any other possible scenario.

Amongst all of the possible scenarios of which a perfect logician would immediately become aware, is one wherein they all merely start counting in the fashion that I was asking about. They would all realize, before the guru said anything, that if there was anything that could allow them to all begin that counting sequence they would be able to deduce their color. So each would then immediately deduce whether there was such a common situation.

All of that would be taking place before the echo of the rule being mentioned had faded. And since even I can figure out how to deduce such a thing without the guru, it is absurd to think they such perfect logicians could not.
Clarify, Verify, Instill, and Reinforce the Perception of Hopes and Threats unto Anentropic Harmony
Else
From THIS age of sleep, Homo-sapien shall never awake.

The Wise gather together to help one another in EVERY aspect of living.

You are always more insecure than you think, just not by what you think.
The only absolute certainty is formed by the absolute lack of alternatives.
It is not merely "do what works", but "to accomplish what purpose in what time frame at what cost".
As long as the authority is secretive, the population will be subjugated.

Amid the lack of certainty, put faith in the wiser to believe.
Devil's Motto: Make it look good, safe, innocent, and wise.. until it is too late to choose otherwise.

The Real God ≡ The reason/cause for the Universe being what it is = "The situation cannot be what it is and also remain as it is".
.
James S Saint
ILP Legend

Posts: 25976
Joined: Sun Apr 18, 2010 8:05 pm

Re: Math Fun

The counting thing doesn't work; it relies on "counting" actually being a hand-wavey introduction of the exact same information that the guru introduces. The only way any deduction is possible from it is if we assume that where they start "counting" is where they have a base of common knowledge, i.e. they know that if there were only X people on the island, they would know that there are X people on the island with blue eyes. But, as I pointed out to Fixed Cross above, if there were X people on the island, they wouldn't be able to learn that there are X people on the island with blue eyes merely by counting the blue eyes they see. This is what Silhouette has been talking about for a long time about taking knowledge out of context.

Also, even if the counting thing did work, they could not magically start. There is no logically best number at which to start counting, so the islanders' flawless logic could not lead them to a common starting point without communicating, which is explicitly excluded in the problem statement.

"Perfect logician" is a potentially ambiguous term, but fortunately the term is defined in the problem statement: "They are all perfect logicians -- if a conclusion can be logically deduced, they will do it instantly." The phrase following the dash should be read as a definition, and no other attributes should be ascribed to a perfect logician.

James, are you arguing that you can prove what the starting number is by deduction? If you think you can, please do. Otherwise, you're making the argument that while we don't know, a perfect logician would know. But you must agree that perfect logician is neither omniscient nor a mind reader, correct? Are we returning to a presumption that every islander wants to leave the island? Isn't it clear that there are simply too many unknowns preventing a perfect islander from deducing a common starting number?

On a tangent, for anyone still following that accepts that solution as the solution: One thing that's quite fascinating about resistance to this problem is the way it happens regularly around a certain threshold: people are generally willing to accept that on an island with 1, 2, or 3 people, the islanders learn something useful from the guru. But at 4, people refuse to follow the same reasoning, and they struggle to come up with reasons. It happens at the other end, too, such that 100, 99, and 98 people can't be expected to act without the intervention of the guru, but 97 people can. But there's a tempting just-so explanation in evolutionary psychology that I'm inclined to accept: that we actually deal with nested knowledge in social situations, but only to a very shallow depth. We might wonder what our parents know, or what our parents know that we know, but rarely will we consider what our parents know that we know that our parents know. It starts to get head-achy around that threshold because our brains aren't built for processing it, and they aren't built that way because it's only in rare and contrived situations when it matters.
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Carleas
Magister Ludi

Posts: 6126
Joined: Wed Feb 02, 2005 8:10 pm
Location: Washington DC, USA

Re: Math Fun

Carleas, you don't know if it works or doesn't.
You refuse to discuss it.
Clarify, Verify, Instill, and Reinforce the Perception of Hopes and Threats unto Anentropic Harmony
Else
From THIS age of sleep, Homo-sapien shall never awake.

The Wise gather together to help one another in EVERY aspect of living.

You are always more insecure than you think, just not by what you think.
The only absolute certainty is formed by the absolute lack of alternatives.
It is not merely "do what works", but "to accomplish what purpose in what time frame at what cost".
As long as the authority is secretive, the population will be subjugated.

Amid the lack of certainty, put faith in the wiser to believe.
Devil's Motto: Make it look good, safe, innocent, and wise.. until it is too late to choose otherwise.

The Real God ≡ The reason/cause for the Universe being what it is = "The situation cannot be what it is and also remain as it is".
.
James S Saint
ILP Legend

Posts: 25976
Joined: Sun Apr 18, 2010 8:05 pm

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